scholarly journals Morse–Bott theory on posets and a homological Lusternik–Schnirelmann theorem

2021 ◽  
pp. 1-24
Author(s):  
D. Fernández-Ternero ◽  
E. Macías-Virgós ◽  
D. Mosquera-Lois ◽  
J. A. Vilches
Keyword(s):  
Morse Theory ◽  
Lusternik Schnirelmann

We develop Morse–Bott theory on posets, generalizing both discrete Morse–Bott theory for regular complexes and Morse theory on posets. Moreover, we prove a Lusternik–Schnirelmann theorem for general matchings on posets, in particular, for Morse–Bott functions.

scholarly journals Strong Discrete Morse Theory and Simplicial L–S Category: A Discrete Version of the Lusternik–Schnirelmann Theorem

2019 ◽  
Vol 63 (3) ◽  
pp. 607-623
Author(s):  
Desamparados Fernández-Ternero ◽  
Enrique Macías-Virgós ◽  
Nicholas A. Scoville ◽  
José Antonio Vilches
Keyword(s):  
Morse Theory ◽  
Discrete Morse Theory ◽  
Discrete Version ◽  
Lusternik Schnirelmann

Morse Theory and the Lusternik–Schnirelmann Category of Quaternionic Grassmannians

2016 ◽  
Vol 60 (2) ◽  
pp. 441-449 ◽  
Author(s):  
Enrique Macías-Virgós ◽  
María José Pereira-sáez ◽  
Daniel Tanré
Keyword(s):  
Morse Theory ◽  
Lusternik Schnirelmann ◽  
Image Position

AbstractThe Lusternik–Schnirelmann category of the quaternionic Grassmannianis known to bek(n − k). In this paper we show that this result can be deduced from Morse theory.

scholarly journals On the Pełczyński conjecture on Auerbach bases

2017 ◽  
Vol 19 (06) ◽  
pp. 1750016 ◽  
Author(s):  
Andrzej Weber ◽  
Michał Wojciechowski
Keyword(s):  
Banach Spaces ◽  
Morse Theory ◽  
Flag Variety ◽  
Dimension Formula ◽  
Lusternik Schnirelmann

We consider Auerbach bases in Banach spaces of dimension [Formula: see text]. We show that there exist at least [Formula: see text] such bases. This estimate follows from the calculation of the Lusternik–Schnirelmann category of the flag variety. A better estimate is obtained for generic smooth Banach spaces using Morse theory.

scholarly journals Fermat Metrics

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1422
Author(s):  
Antonio Masiello
Keyword(s):  
Variational Methods ◽  
Morse Theory ◽  
Sagnac Effect ◽  
Variational Theory ◽  
Global Hyperbolicity ◽  
Fermat Principle ◽  
Multiple Image ◽  
Light Rays ◽  
Stationary Spacetimes

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.

scholarly journals Topology of Subvarieties of Complex Semi-abelian Varieties

2020 ◽  
Author(s):  
Yongqiang Liu ◽  
Laurentiu Maxim ◽  
Botong Wang
Keyword(s):  
Euler Characteristic ◽  
Morse Theory ◽  
Characteristic Property ◽  
Abelian Varieties ◽  
Betti Numbers ◽  
Local Systems ◽  
Affine Manifolds ◽  
Cyclic Covers ◽  
Rank One ◽  
Generic Vanishing

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

Periodic Problems with the Scalar p-Laplacian Resonant at any Eigenvalue via Critical Point Methods

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Donal O’ Regan ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Critical Point ◽  
Morse Theory ◽  
Critical Point Theory ◽  
Smooth Solution ◽  
Point Theory ◽  
Reaction Term ◽  
Variational Techniques ◽  
Periodic Problems

AbstractWe consider nonlinear periodic problems driven by the scalar p-Laplacian with a Carathéodory reaction term. Under conditions which permit resonance at infinity with respect to any eigenvalue, we show that the problem has a nontrivial smooth solution. Our approach combines variational techniques based on critical point theory with Morse theory.

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